Dic₂₆₇₆ #72212899

order	10704
automorphism	Aut(G) order 9505152
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₂₆₇₆

Splittings

Hall	C₃×C₂₂₃ ⋊ Q₁₆
Sylow	C₂₂₃ ⋊ Dic₁₂
Sylow	C₃ ⋊ Dic₈₉₂

Subgroups

	subgroup	G/H
center	C₂	D₂₆₇₆
commutator, cosocle	C₃×C₄×C₂₂₃	C₂²
Frattini	C₄	C₂×D₆₆₉
Fitting	C₃×C₈×C₂₂₃	C₂
socle, prime radical	C₂×C₃×C₂₂₃	D₄
hypercenter	C₈	D₆₆₉
hypocenter	C₃×C₂₂₃	Q₁₆
LCS	G ▷ C₃×C₄×C₂₂₃ ▷ C₂×C₃×C₂₂₃ ▷ C₃×C₂₂₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 242
transitive degree	μ_t(G) = 10704

A minimal faithful permutation representation — 2 generators on 242 points:

(1,2,3,4,5,6,7,8)(9,16,15,14,13,12,11,10)(17,18,19)(20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242);(1,9,5,13)(2,10,6,14)(3,11,7,15)(4,12,8,16)(18,19)(21,242)(22,241)(23,240)(24,239)(25,238)(26,237)(27,236)(28,235)(29,234)(30,233)(31,232)(32,231)(33,230)(34,229)(35,228)(36,227)(37,226)(38,225)(39,224)(40,223)(41,222)(42,221)(43,220)(44,219)(45,218)(46,217)(47,216)(48,215)(49,214)(50,213)(51,212)(52,211)(53,210)(54,209)(55,208)(56,207)(57,206)(58,205)(59,204)(60,203)(61,202)(62,201)(63,200)(64,199)(65,198)(66,197)(67,196)(68,195)(69,194)(70,193)(71,192)(72,191)(73,190)(74,189)(75,188)(76,187)(77,186)(78,185)(79,184)(80,183)(81,182)(82,181)(83,180)(84,179)(85,178)(86,177)(87,176)(88,175)(89,174)(90,173)(91,172)(92,171)(93,170)(94,169)(95,168)(96,167)(97,166)(98,165)(99,164)(100,163)(101,162)(102,161)(103,160)(104,159)(105,158)(106,157)(107,156)(108,155)(109,154)(110,153)(111,152)(112,151)(113,150)(114,149)(115,148)(116,147)(117,146)(118,145)(119,144)(120,143)(121,142)(122,141)(123,140)(124,139)(125,138)(126,137)(127,136)(128,135)(129,134)(130,133)(131,132)

Polycyclic presentation:

2^4,3,223|a^2=d,b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^2,f^a=f^222

Conjugacy classes & character table

2679 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₅₃₅₂, Dic₈₀₂₈, Dic₁₃₃₈₀
maximal normal subgroup of	Dic₅₃₅₂
maximal quotient of	Dic₈₀₂₈, Dic₁₃₃₈₀

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 3, 223]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.6^222);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic2676 #72212899